Set Theory Problem: equality of functions

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Homework Statement

Prove:

Let F and G be functions. F = G if and only if domF = domG and F(x) = G(x) for all x \in domF.

Homework Equations

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The Attempt at a Solution

If F = G, then (xFy if and only if xGy) (Substitutivity of identicals)

If (xFy if and only if xGy), then F = G (Axiom of extensionality)

So F = G if and only if (xFy if and only if xGy)

Since F and G are functions,

For all x \in domF, xFy if and only if y = F(x)
For all x \in domG, xGy if and only if y = G(x)

xFy if and only if x \in domF and xFy
xGy if and only if x \in domG and xGy

xFy if and only if x \in domF and y = F(x)
xGy if and only if x \in domG and y = G(x)

So

F = G if and only if ((x \in domF and y = F(x)) if and only if (x \in domG and y = G(x)))

which is equivalent to

F = G if and only if ((x \in domF and x \in domG and y = F(x) and y = G(x)) or (x \in domF and x \in domG and y \neq F(x) and y \neq G(x)) or (x \notin domF and x \notin domG) or (x \in domF and x \notin domG and y \neq F(x)) or (x \notin domF and x \in domG and y \neq G(x))

I was hoping I could show that domF = domG and F(x) = G(x) for all x \in domF was equivalent to this but I couldn't. This is as close as I could get:

(domF = domG and F(x) = G(x) for all x \in domF) if and only if (x \in domF if and only if x \in domG and, if y \in domF, then y = F(x) if and only if y = G(x))

(domF = domG and F(x) = G(x) for all x \in domF) if and only if (((x \in domF and x \in domG and y = F(x) and y = G(x)) or (x \in domF and x \in domG and y \neq F(x) and y \neq G(x)) or (x \notin domF and x \notin domG))

This is close to what I had above but not quite.

I don't know. I think I need to get a set theory book that has solutions; I am currently working through "Introduction to Set Theory" by Hrbacek and Jech. I might switch to "Naive Set Theory" by Paul Hamos; it doesn't even have exercises I don't think.

Any help would be appreciated.

 

[micromass]

May I ask you how you defined "function"? Is this an exercise of Hrbacek and Jech, if so, which one? I need to see all the definitions before I can answer, since a function is probably something different for me...

On a related note, Hrbacek and Jech is a very good book, I really recommend it. Halmos is more about naive set theory, and is less formal. If you want to learn modern set theory, then Hrbacek and Jech is the best. If you just want to use the set theory somewhere, then I guess Halmos is easier...

Also I would like to add this reference: staff.science.uva.nl/~vervoort/AST/ast.pdf The theory in there is not at all easy, but it has some good and fun exercises (with solutions)

 

[micromass]

Dont mind the above post, I checked Hrbacek and Jech and found it... For the record, I don't really like that definition...

So you indeed have

F=G~\text{iff}~ (xFy ~\Leftrightarrow~xGy)

The domain of F is

dom(F)=\{x~\vert~\exists y:~xFy\}=\{x~\vert~\exists y:~xGy\}=dom(G)

So the domains are equal. Now take an x in dom(F). Then there exists a y such that xFy=xGy. This means exactly that F(x)=G(x) (since y is unique).